HIERARCHY OF INTEGRABLE GEODESIC FLOWS
Peter Topalov Abstract
A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics.
1. Introduction
In papers [1], [2], [3] a simple approach was suggested for obtaining first integrals of Hamiltonian systems if a trajectorial diffeomorphism is given. This approach is closely related to the ideas presented in [4].
Recall briefly the main construction (for details see [1], [2]). Let v and ¯v be Hamiltonian systems on the symplectic manifolds (M2n, ω) and ( ¯M2n, ¯ω) with Hamiltonians H and ¯H respectively. Consider the isoenergy surfaces
Qdef= {x ∈ M2n| H(x) = h}, ¯Qdef= {x ∈ ¯M2n| ¯H(x) = ¯h}, where h and ¯h are regular values of the functions H and ¯H.
Definition 1. A diffeomorphism φ : Q→ ¯Q is said to be trajectorial, if it takes the trajectories of the system v to the trajectories of the system ¯v.
Let φ : Q → ¯Q be a trajectorial diffeomorphism. Let us denote the restrictions ω|Q and ¯ω|Q¯ also by the letters ω and ¯ω. Then the pull- back vanish (of course, also Lv¯ω = 0). It is obvious that the kernels of the forms ω and φ∗ω coincide with the linear span of the vector v.¯ Therefore, these forms induce two non-degenerate tensor fields on the quotient bundle T Q/ v. Thus, the characteristic polynomial of the operator ω−1◦ φ∗ω is preserved by the flow v.¯
In papers [2], [3] the construction was applied to a classical example where such a diffeomorphism exists —the geodesic flows corresponding to a pair of geodesically equivalent metrics.
Let g and ¯g be Riemannian metrics on the manifold Mn.
Definition 2. The pseudo-Riemannian metrics g and ¯g are called geo- desically equivalent iff they have the same geodesics (considered as un- parameterized curves).
For geodesically equivalent Riemannian metrics, a trajectorial diffeo- morphism Φ is given by the formula Φ : (x, ξ) →
x,||ξ||||ξ||g
¯ gξ
, where ξ∈ TxMn.
Denote by Mq(g, ¯g) (1 ≤ q ≤ n) the set of points y ∈ Mn such that the Riemannian metrics g and ¯g have exactly q distinct eigenvalues in an open neighborhood of the point y. If the set Mn is everywhere dense in Mn we say that the Riemannian metrics g and ¯g are strictly non-proportional on Mn.
In the papers [2], [3] the following was proved.
Theorem 1. Suppose the Riemannian metrics g and ¯g are geodesically equivalent; then
a) the geodesic flow of the metric g admits n integrals (k≥ 0) Ikdef= (−1)k
det(g) det(¯g)
k+2n+1
¯
g(Skv, v), (1)
where Sk
def= ¯Gk− σ1( ¯G) ¯Gk−1+· · · + (−1)kσk( ¯G) (2)
and ¯Gdef= (gij¯gjk), σi( ¯G) are the elementary symmetric polynomi- als of degree i;
b) the integrals Ik (k≥ 0) are in involution1;
c) ifMq = ∅, then the rank of the differentials dI0, . . . , dIn−1 is equal to q almost everywhere inT Mq.Moreover, onT Mq we have
rk(dI0, . . . , dIn−1) = rk(dI0, . . . , dIq−1)≤ q.
(3)
This theorem is closely related to some results proved by U. Dini, P. Painlev´e, T. Levi-Civita and R. Liouville (see [5], [6]).
In the present paper we assign to any pair of geodesically equivalent metrics a hierarchy of integrable geodesic flows (Theorem 5). The paper is organized as follows.
In Section 2 we present some important facts from the theory of geo- desic mappings needed for the sequel.
In Section 3 we assign to any pair of strictly non-proportional geo- desically equivalent metrics g and ¯g a family of completely integrable
1I.e., makingthe Legendre transformation we obtain n functions commutingwith respect to the canonical symplectic structure onT∗M .
Riemannian metricsS(g, ¯g) (see Theorem 5). Some simple properties of these families are obtained.
In Section 4 the results obtained in Section 3 are applied to the metrics
dg2 def=
n i=1
dxi2 (4)
and
d¯g2 def= 1
n i=1
xi ai
2
n i=1
dxi2 ai (5)
which are geodesically equivalent on the standard ellipsoid Eh def
n =
i=1xi 2 ai = h
, h > 0. We give explicit formulae for the family S(dg2|Eh, d¯g2|Eh) in terms of some metrics on the whole Rn (Theo- rem 6). It is interesting that this family contains the metric on the Poisson sphere, well-known in classical mechanics. It immediately per- mits us to prove that the geodesic flow on the standard ellipsoid and the geodesic flow on the Poisson sphere have the same Liouville foliation (Corollary 3). This theorem is a multidimensional generalization of the well-known one in the two-dimensional case. In addition, we give an explicit formula for a metric geodesically equivalent to the metric on the Poisson sphere (Corollary 2).
In Section 5 we obtain a family of completely integrable Hamiltonians with respect to the standard Lie-Poisson bracket on e(3)∗(the dual space to the Lie algebra e(3)). In particular, we obtain an algebraic Hamil- tonian H(1)such that the corresponding Hamiltonian system sgrad H(1) is orbitally equivalent to the Euler case of the free motion of the rigid body (see Corollary 6).
Throughout the paper the standard agreement holds, i.e. Riemannian metric means a positive-definite symmetric form and pseudo-Riemannian metric means a non-degenerate symmetric form. We consider mainly Riemannian metrics; nevertheless many results hold in the pseudo- Riemannian case.
The author is grateful to A. V. Bolsinov, A. T. Fomenko, V. V. Kozlov, V. S. Matveev, S. Tabachnikov and I. A. Taimanov for useful discussions.
This paper was written during my stay at the Max-Planck-Institut f¨ur Mathematik in Bonn. It is pleasure to thank the Institut for its hospital- ity and financial support. The author is partially supported by MESC grant MM-810/98.
2. Geodesically equivalent metrics and the corresponding 1-parameter family The following lemma is needed for the sequel.
Lemma 1 (see [8]).
1) Suppose the pseudo-Riemannian metrics g and ¯g are geodesically equivalent; then the tensors aij and λi
aijdef
= Aαigαj, (6)
λidef= −Aαiψα, (7)
where (n + 1)ψi = 12∂ilngg¯, g = det(gij), and the operator A is given by formula
Aij(g, ¯g)def= g¯ g
1 n+1
¯ giαgαj, (8)
satisfy the equation
aij,k= λigjk+ λjgik. (9)
Here aij,k denotes the covariant derivative ∇kaij, where ∇ is the Levi-Civita connection corresponding to the metric g.
2) Conversely, if a non-degenerate symmetric tensor field aij and an 1-form λi satisfy equation (9), then the metric
¯ gij def=
gˆ g
ˆ gij, (10)
where ˆgijdef= giαaαβgβj, is geodesically equivalent to g.
It is obvious that A is self-adjoint with respect the both metrics.
Now, using Lemma 1 we can prove
Proposition 1. Suppose the pseudo-Riemannian metrics g and ¯g are geodesically equivalent; then if for some parameters α and β the opera- tor (αA + β) is invertible on Mn, then the pseudo-Riemannian metrics g and
¯
gα,β(X, Y )def= 1
det(αA + β)g((αA + β)−1X, Y ), (11)
are geodesically equivalent.
Remark 1. Remark here that ¯g0,1= g and ¯g1,0= ¯g.
Remark 2. If the manifold Mn is compact, then formula (11) gives a 1-parameter family of geodesically equivalent metrics. Moreover, in a small neighborhood of each point of Mn we also obtain 1-parameter family.
Remark 3. Let us take two representatives ¯gα,β and ¯gλ,µ. Now, we are able to apply Proposition 1 again. The corresponding family of geodes- ically equivalent metrics is
¯
ga,b = 1
det((aλ + bα)A + (aµ + bβ))g((aλ + bα)A + (aµ + bβ))−1. (12)
Therefore, if (α : β)= (λ : µ), then we obtain the same family. Remark here also that
A(¯gα,β, ¯gλ,µ) = λA + µ αA + β. (13)
Proof of Proposition 1: Let g and ¯g be geodesically equivalent metrics.
Using (6), we obtain that the symmetric form a = gA satisfies equa- tion (9) for some λ. Hence, the form ac
def= gA + cg (c = const) also satisfied this equation. We use here that∇ is the Levi-Civita connection corresponding to the metric g. Assume that ac is non-degenerate on Mn. We get ˆg def= g(gA + cg)−1g = g(A + c)−1 and using the converse formula (10) obtain that the pseudo-Riemannian metric
¯
gc= 1
det(A + c)g(A + c)−1 (14)
is geodesically equivalent to g.
Now, using Proposition 1 we are able to present another ‘non-sym- plectic’ proof of the first item of Theorem 1.
Corollary 1. If the Riemannian metrics g and ¯g are geodesically equiv- alent, then the geodesic flow of the metric g admits a 1-parameter family of first integrals
Iα,β(g, ¯g)def= det(αA + β)g(αA + β)−1. (15)
Proof of Corollary 1: The corollary easily follows from the next theorem (see [6]).
Theorem 2 (Painlev´e). If the metrics g and ¯g are geodesically equiva- lent, then the function
I0(g)def=
det(g) det(¯g)
n+12
¯ g (16)
is an integral of the the geodesic flow of the metric g.
In a small neighborhood of each point of Mnwe can find a 1-parameter family ¯g1,β of geodesically equivalent Riemannian metrics. Therefore, locally the Riemannian metrics g and ¯g1,β are geodesically equivalent, where β∈ (a, b), a < b. It permits us to apply Theorem 2. We have
I1,β(g)def=
det(g) det(¯g1,β)
n+12
¯ g1,β (17)
= det(A + β)g(A + β)−1 (18)
= I0+ I1β +· · · + In−1βn−1, (19)
where the functions Ikare the same as in Theorem 1. The last equality in this chain can easily be proved. Therefore, the functions Ik are integrals in a small neighborhood of each point of Mn. But they are globally defined on Mn. Thus, they are integrals.
Remark 4. If we apply Corollary 1 to the geodesically equivalent met- rics g and ¯gα,β ((α : β)= (0 : 1)), then we’ll obtain the same family of integrals.
Remark 5. Let us consider the map Φ : ξ →||ξ||||ξ||gg¯ξ2, Φ : (T M)0→ (T M)0. (20)
It is obvious that Φ maps each geodesic trajectory of the metric g into a geodesic trajectory of the metric ¯g. Therefore, the pull-back Φ∗(Iα,β(¯g, g)) gives a family of integrals for the geodesic flow of the metric g. It can easily be checked that
Φ∗(Iα,β(¯g, g)) = g(ξ, ξ)
I1,0(g, ¯g)Iβ,α(g, ¯g).
(21)
Thus, we won’t be able to obtain new family of integrals.
2Sometimes we will denote this mappingby Φ(g, ¯g).
3. Hierarchy of integrable flows
As we have seen if the Riemannian metrics g and ¯g are geodesically equivalent, then they are contained in a family of geodesically equivalent metrics
¯
gα,β= 1
det(αA + β)g(αA + β)−1, (22)
where α and β are parameters.
Remark 6. In the following sections we consider only Riemannian met- rics although the most of the constructions pass in pseudo-Riemannian case.
It is interesting that using geodesically equivalent Riemannian met- rics g and ¯g we can produce another families of geodesically equivalent metrics. We need
Theorem 3 (see [8]). If the Riemannian metrics g and ¯g are geode- sically equivalent, then for each integer k the Riemannian metrics g(k) def= gAk and ¯g(k) def= ¯gAk are also geodesically equivalent.
This theorem may be proved by direct calculations. For details see paper [8]. We will slightly generalize this result later using some other arguments. At first we need some notations.
Let B be a self-adjoint operator on the connected Riemannian mani- fold (Mn, g). By definition, put
r(B)def= inf
x∈Mmin{spec B(x)}
(23)
and
R(B)def= sup
x∈Mmax{spec B(x)}.
(24)
Denote by I the set obtained by adding to the interval (r, R) its endpoints iff they are achieved for some x ∈ Mn. Let us consider the set of all real Laurent series La(x) =
kck(x− a)k which are convergent on some open neighborhood of I. Denote by ω+(B) the cone of all finite linear combinations of such series which give positive functions on I. Of course, we are able to consider more general set of functions but it will only complicate our construction.
Theorem 4. Suppose the Riemannian metrics g and ¯g are geodesically equivalent and F (x)∈ ω+(A); then the metrics gF def= gF (A) and ¯gF def=
¯
gF (A) are also geodesically equivalent.These metrics are contained in a family of geodesically equivalent Riemannian metrics
¯
gα,βF def= 1
det(αA + β)gF (A)(αA + β)−1, (25)
where α and β are parameters such that the operator (αA + β) is non- degenerate.
Proof of Theorem 4: Let g and ¯g be geodesically equivalent Riemannian metrics on a manifold Mn. Denote by ρ1, . . . , ρm(1≤ m ≤ n) the com- mon eigenvalues of the metrics g and ¯g. Suppose the functions ρ1, . . . , ρm are different at every point of an open domainD ⊂ Mn. In the paper [6], T. Levi-Civita proved that for every point P ∈ D there is an open neigh- borhoodU(P ) ⊂ D and a coordinate system ¯x = (¯x1, . . . , ¯xm) (inU(P )), where ¯xi= (x1i, . . . , xkii), (1≤ i ≤ m), such that the quadratic forms of the metrics g and ¯g have the following form:
g( ˙¯x, ˙¯x) =
m i=1
Πi(¯x)Ai(¯xi, ˙¯xi), (26)
¯
g( ˙¯x, ˙¯x) =
m i=1
ρiΠi(¯x)Ai(¯xi, ˙¯xi), (27)
where Ai(¯xi, ˙¯xi) are positive-definite quadratic forms in the velocities ˙¯xi
with coefficients depending on ¯xi, Πidef
= (φi− φ1) . . . (φi− φi−1)(φi+1− φi) . . . (φm− φi), (28)
ρi= 1 φ1. . . φm
1 φi (29)
and φ1, φ2, . . . , φm(0 < φ1< φ2<· · · < φm) are smooth functions such that
φi=
φi(¯xi), if ki= 1 constant, else.
Definition 3. Let metrics g and ¯g be given by formulae (26) and (27) in a coordinate chartU. Then we say that the metrics g and ¯g have Levi- Civita local form of (type m), and the coordinate chartU is Levi-Civita coordinate chart (with respect to the metrics).
In the paper [6], Levi-Civita proved that the metrics g and ¯g given by formulae (26) and (27) are geodesically equivalent.
Denote byM the set of all point in Mn which are contained in some Levi-Civita chart, i.e. x ∈ M iff there is a Levi-Civita chart U which contains x. By definition, M is an open subset of Mn. In [2], [3] was proved thatM is everywhere dense in Mn.
In every Levi-Civita coordinate chart we have
g¯ g
n+11
=
1 φk11−1. . . φkmm−1
1
φ1. . . φm = const. 1 φ1. . . φm, (30)
(¯gikgkj) = diag
1
ρ1, . . . , 1 ρ1
k1
; . . . ; 1
ρm, . . . , 1 ρm
km
. (31)
Therefore,
A(g, ¯g) = const. diag(φ1, . . . , φ1
k1
; . . . ; φm, . . . , φm
km
).
(32)
Hence,
g(F (A) ˙¯x, ˙¯x) =
m i=1
Πi(¯x)Ai(¯xi, ˙¯xi), (33)
and
¯
g(F (A) ˙¯x, ˙¯x) =
m i=1
ρiΠi(¯x)Ai(¯xi, ˙¯xi), (34)
where Ai = fi(¯xi)Ai and fi(¯xi) is a smooth function depending of the variable ¯xi. Therefore, in any Levi-Civita chart the metrics gF (A) and
¯
gF (A) are geodesically equivalent. Let us consider the map Φ : ξ →
||ξ||g
||ξ||g¯ξ,
Φ : (T M)0→ (T M)0. (35)
Denote by v and ¯v the vector fields corresponding to the geodesic flows of the metrics gF (A) and ¯gF (A). Therefore, the smooth vector fields ¯v= 0 and Φ∗(v) = 0 are proportional on a dense subset of the iso-energetic surface UrM , (r > 0). Thus, their directions coincide everywhere. It yields that the metrics gF (A) and ¯gF (A) are geodesically equivalent.
Finally, using Proposition 1, we obtain a family of geodesically equiv- alent metrics
¯
gα,βF def= 1
det(αA(gF, ¯gF) + β)gF(αA(gF, ¯gF) + β)−1 (36)
= 1
det(αA + β)gF(αA + β)−1. (37)
This completes the proof of Theorem 4.
If the Riemannian metrics g and ¯g are strictly non-proportional geo- desically equivalent on Mn, then the metrics gF and ¯gF are also strictly non-proportional geodesically equivalent. Therefore, using Theorem 1, we obtain that all Riemannian metrics ¯gFα,β are completely integrable.
Theorem 5. If g and ¯g are strictly non-proportional geodesically equiv- alent Riemannian metrics, then the family of Riemannian metrics
S(g, ¯g)def= {¯gFα,β| F (x) ∈ ω+(A), α, β are parameters} (38)
consists of completely integrable Riemannian metrics.
Remark 7. An explicit formula for the integrals of the metric gF is Iα,β(gF, ¯gF) = det(αA + β)gF (A)(αA + β)−1
(39)
=
g
¯ g
n+12
Iβ,α(¯gF, gF).
(40)
Definition 4. The family S(g, ¯g) is called S-hierarchy of integrable Rie- mannian metrics corresponding to the pair g and ¯g.
Remark 8. The family S(g, ¯g) in some sense is ‘complete’. It means that if we take two representatives ¯gFα,β and ¯gFλ,µ ((α : β) = (λ : µ)), then S(¯gα,βF , ¯gλ,µF ) = S(g, ¯g).
Now, we are going to establish some relations between the flows of the metrics of a given S-hierarchy.
Proposition 2. The metrics from a given S-hierarchy have identical
‘Liouville foliations’.It means that for every two metrics g1 and g2 we can find
(i) n independent almost everywhere on (T M)0 pairwise commuting integrals {Ik(1)}nk=0−1 for the geodesic flow of the metric g1;
(ii) n independent almost everywhere on (T M)0 pairwise commuting integrals {Ik(2)}n−1k=0 for the geodesic flow of the metric g2;
(iii) a diffeomorphism φ : (T M)0→ (T M)0, such that φ∗Ik(2) = Ik(1).
Proof of Proposition 2: Suppose g1, g2∈ S(g, ¯g). Without loss of gener- ality it can be assumed that g1= g and g2= ¯gGα,β. Let us consider the commutative diagram
(T M)0 ✛Φ (T M)0
❦◗◗
◗◗◗ (T M)0
✻Ψ ,
where Φ def= Φ(gG, ¯gα,βG ) and Ψ : ξ → (
G(A))−1ξ. Let φ def= Φ◦ Ψ.
Now, the proposition easily follows from Remark 5 and Remark 7.
Let g and ¯g be strictly non-proportional geodesically equivalent Rie- mannian metrics on the manifold Mn. Consider the diagram
g(k)
❄✛g.e.✲ ¯g(k)❄
g(k+1)
❄ ✛g.e.✲ ¯g(k+1)❄
g(k+2)
❄ ✛g.e.✲ ¯g(k+2)❄
❄ ❄
,
where the horizontal arrows mean that the corresponding Riemannian metrics are geodesically equivalent. Recall that g(k+1) = g(k)A, where
A def= A(g, ¯g). We proved that all metrics of the above sequence are completely integrable.
Proposition 3. Let us consider the Riemannian metrics g(k)and ¯g(k+2). Making the corresponding Legendre transformations we obtain two Hamiltonian systems on the cotangent bundleT∗M which are completely integrable and the corresponding integrals, canonically given by Theo- rem 1, coincide.
Proof of Proposition 3: Making the Legendre transformations we obtain two families of pairwise commuting functions onT∗M
a) Iα,β(g(k), ¯g(k)) = det(αA + β)(αA + β)−1(gAk)−1; b) Iα,β(¯g(k), g(k)) = det(αA−1+ β)(αA−1+ β)−1(¯gAk)−1. We obviously have
Iα,β(¯g(k), g(k)) =|α + βA|
|A|
(αA−1+ β)−1A−1
A−k+2
A−1g¯−1
=
1
|A|
g
¯ g
n+11
(det(α + βA)(α + βA)−1A−k+2g−1
= Iβ,α(g(k−2), ¯g(k−2)).
This completes the proof.
4. S-hierarchy for the ellipsoid In [2] was proved that the restrictions of the metrics
dg2 def=
n i=1
dxi2 (41)
and
d¯g2 def= 1
n i=1
xi ai
2
n i=1
dxi2 ai (42)
to the ellipsoid Eh
def= n i=1 xi 2
ai = h
, h > 0 are geodesically equi- valent3. Moreover, these metrics are strictly non-proportional on the ellipsoid. Without loss of generality we can assume that a1< a2<· · · <
an.
3The same result was independently obtained by S. Tabachnikov in [9].
Our aim in the present section is to find explicit formulae for S- hierarchy S(g, ¯g). Moreover, we intend to extend these metrics on the whole Rn in a natural manner.
By definition, put Adef= diag(a1, . . . , an). Denote by., . the natural pairing in Rn.
Let us consider two smooth operators on Rn\{0}
A(x)def= A− 1
A−1x, xx⊗ x + (A−1x)⊗ (A−1x) (43)
and
A(x)def= A−1−(A−1x)⊗ (A−2x) + (A−2x)⊗ (A−1x)
A−1x, A−1x (44)
where x = (x1, . . . , xn).
Proposition 4. The operators A and A satisfy the following condi- tions:
1) both of them are self-adjoint with respect to the metric g;
2) for each x∈ Rn the tangent space TxEh is invariant subspace for these operators;
3) A(x)(A−1x) =
A−1x, A−1x
A−1x and A(x)(A−1x) =−
A−2x, A−1x
A−1x, A−1xA−1x;
4) the restriction of the operator A(x) to each ellipsoid Eh, h > 0 coincides (up to multiplication on a positive constant) with the operator A(g|Eh, ¯g|Eh) and the restriction of A coincides (up to multiplication on a positive constant) with [A(g|Eh, ¯g|Eh)]−1. These conditions determine the operators A and A uniquely.
Proof of Proposition 4: Items 1) and 3) are obvious.
Let us prove 2). Suppose that ξ∈ TxEh, i.e.
A−1x, ξ
= 0. We have
A−1x,A(x)(ξ)
=
A−1x, Aξ− x, ξ
A−1x, xx
=x, ξ − x, ξ = 0 and
A−1x,A(x)(ξ)
=
A−1x, A−1ξ−
A−2x, ξ
A−1x, A−1xA−1x
=
A−1x, A−1ξ
−
A−2x, ξ
= 0.
Before proving 4) we need the next simple lemma.
Lemma 2. Let V be an oriented real vector space and T ⊂ V be an oriented hyperplane.If g and ¯g are positive defined metrics, then
vol(¯g|T)
vol(g|T) =vol(¯g)
vol(g)g(ng, n¯g), (45)
where ng and ng¯ are positive directed unit normals to the subspace T calculated for the metrics g and ¯g correspondingly.
The proof of Lemma 2 is trivial.
Now, using the lemma, we obtain
det(¯g|Tx) det(g|Tx)
(n−1)+11
=
A−1x, x1n
(Πni=1ai)n1
1
A−1x, A−1x. (46)
If ξ, η∈ TxEh, then we get
¯
gx(A(x)(ξ), η) = 1
A−1x, A−1x
ξ− x, ξ
A−1x, xA−1x, η
= 1
A−1x, A−1xξ, η = gx(ξ, η)
A−1x, A−1x. Therefore, we have proved that
A(x)|TxEh =
Πni=1ai h
n1
A(g|Eh, ¯g|Eh).
(47)
Similarly, we have
g(A(x)(ξ), η) =
A−1ξ−
A−2x, ξ
A−1x, A−1xA−1x, η
=
A−1ξ, η
=
A−1x, A−1x
¯ g(ξ, η).
Hence,
A(x)|TxEh =
h
Πni=1ai
n1
[A(g|Eh, ¯g|Eh)]−1. (48)
Finally, it is obvious that if a self-adjoint operator is given on an invariant hyperplane, then the normal vector is an eigenvector and the corresponding eigenvalue determines the operator uniquely on the whole space.
Theorem 6. Consider the family of Riemannian metrics g(k) def=
Ak(x)(ξ), η
and ¯g(k) def= A−1x,A1 −1x
Ak−1(x)(ξ), η
(k ∈ Z) on Rn\{0}.
1) The restriction of g(k) to any ellipsoid Eh gives a completely inte- grable Riemannian metric.The corresponding integrals are given by formula
Iα,β(k) def= det(αA(x) + β)
αA−1x, A−1x + βAk(x)(αA(x) + β)−1, (49)
where α and β are parameters.
2) The restriction of ¯g(k) to any ellipsoid Eh gives a completely inte- grable Riemannian metric.The corresponding integrals are given by formula
I¯α,β(k) def= Iβ,α(k)
A−1x, A−1x2. (50)
3) The metrics g(k)and ¯g(k) are contained in a 1-parameter family of metrics
¯
g(k)α,βdef= α
A−1x, A−1x + β
det(αA(x) + β) A(k)(x)(αA(x) + β)−1, (51)
and the restrictions of any two metrics from this family to the ellipsoid Eh are geodesically equivalent.
The proof easily follows from Proposition 4, Theorem 4 and Theo- rem 5.
The Ellipsoid and the Poisson sphere. In the case k = 1 we obtain that the metrics
dg(1)2=A(x)dx, dx
(52)
=Adx, dx − x, dx2
A−1x, x+
A−1x, dx2
(53) and
d¯g(1)2= 1
A−1x, A−1xdx, dx
(54)
are geodesically equivalent on the ellipsoid Eh, h > 0.
The metric ¯g(1)|Ehcan be considered as a metric on the homogeneous space Sn = SO(n + 1)/SO(n), where the kinetic energy of the (n + 1)- dimensional rigid body is spread left-invariantly on the whole SO(n + 1).
This construction generalize the well-known metric on the Poisson sphere considered in the classical mechanics (see [10]).
Corollary 2. The restriction of the metric dg(1)2 to the ellipsoid Eh has the same geodesic lines as the metric on the Poisson sphere.
Using Proposition 2, we obtain
Corollary 3. The geodesic flows on the ellipsoid and the Poisson sphere have identical Liouville foliations.
If n = 2 the last result is well-known. Moreover, in [11] A. T. Fomenko and A. V. Bolsinov using the theory of orbital equivalence of the inte- grable Hamiltonian systems proved that for every 2-dimensional ellipsoid we can take a suitable Poisson sphere such that the corresponding geo- desic flows are continuously orbitally equivalent. It is interesting to solve the same problem in multidimensional case.
Thestandard sphereand thePoisson sphere. In the case k =−1 we have
dg2(−1)=
A(x)dx, dx
(55)
=
A−1dx, dx
− 2
A−1x, dx
A−2x, dx
A−1x, A−1x . (56)
The last term on the right side vanish on T Eh. Therefore, we can think that dg(2−1) =
A−1dx, dx
. Changing the variables x = √
Ay we see that g(−1) is the standard metric on the sphere. Further,
d¯g2(−1)=
!A−1dx, A−1dx
− A−1x,A−1dx2
A−1x,A−1x
"
A−1x, A−1x . (57)
Corollary 4. The restriction of the metric
d¯g(−1)2 =
A−1dy, dy
− A−1y,dy2
A−1y,y
A−1y, y
(58)
to the sphere Sn ={y, y = h > 0} has the same geodesic lines as the standard metric on the sphere.
Using Proposition 3, we obtain
Corollary 5. The metric on the Poisson sphere d¯g(1)2 |Eh and the stan- dard metric on the sphere dg2(−1)|Eh are completely integrable on T∗M such that the corresponding integrals are the same.
5. A family of integrable Hamiltonians in e(3)∗ It is well-known that the Euler-Poisson equations which describe the motion of the rigid body can be written as Euler equations on the dual space of the Lie algebra e(3) (see [12], [13]). More precisely, if the coordinates in e(3)∗ are denoted by (r1, r2, r3, s1, s2, s3), then the Lie- Poisson bracket can be given by relations{si, sj} = 8ijksk,{ri, rj} = 0, {si, rj} = 8ijkrk. Therefore, if H = H(r, s) is a Hamiltonian function, then the Euler equations take the form of Kirchoff equations
˙s = s×∂H
∂s + r×∂H
∂r, (59)
˙r = r×∂H
∂s. (60)
The Lie-Poisson bracket has two annihilators F1 = r21+ r22+ r23 and F2= r1s1+ r2s2+ r3s3. The Hamiltonian function for the rigid body is
Hdef= 1 2
s21 I1 +s22
I2 +s23 I3
+ mgl, r , (61)
where s = (s1, s2, s3) denotes the angular momentum of the body, l = (l1, l2, l3) denotes the coordinates of the center of gravity of the body, I = diag(I1, I1, I3) is the tensor of inertia, r = (r1, r2, r3) —the coordinates of the unit vertical vector in the space, and mg is the weight of the body.
The cotangent bundle T∗S2 supplied with the canonical symplectic structure dp∧ dq is symplectomorphic to the manifold Oh
def= {F1 = h, F2 = 0} ⊂ e(3)∗ (see [14]). Therefore, any integrable Hamiltonian on T∗S2 gives an integrable Hamiltonian onOh. Moreover, if the first Hamiltonian is polynomial in impulses, then the second one is also poly- nomial in momenta s = (s1, s2, s3) and has the same degree. Using the results of [14] and Lemma 2 it is easy to prove
Lemma 3. Suppose that the Riemannian metric dg2 = gijdxidxj, x = (x1, x2, x3) is smooth in a neighborhood of the sphere Sh2={x, x = h}
and the restriction dg2|Sh2 gives a metric whose geodesic flow is com- pletely integrable; then the Hamiltonian function
Hg(r, s)def=
det E det G
1 e(ne, ng)
i,j
gij(r)sisj, (62)
where de2 = (dx1)2+ (dx2)2+ (dx3)2, G = (gij), E = (eij) and ne and ng are the unit external normal vectors to the sphere Sh2 calculated with respect to the metrics e and g respectively at the point x = r ∈ Sh2, is completely integrable on the submanifold Oh.
In Section 4 we found a family of metrics on Rnsuch that their restric- tion on the ellipsoid Ehgive a family of integrable metrics. Now, using these metrics and Lemma 3 we are able to find a family of integrable Hamiltonians in e(3)∗.
Theorem 7. Consider the operators A(r)def= A−(√
Ar)⊗ (√ Ar)
r, r + (A−12r)⊗ (A−12r) (63)
and
B(k)(r)def= √
AAk(r)√ A.
(64)
For any fixed integer k the Hamiltonians H(k) def= 1
A−1r, rk+1
B(k)s, s
(65)
and
H¯(k) def= 1
A−1r, rk−1
B(k−1)s, s
(66)
are Liouville integrable on any orbit Oh.Moreover, the corresponding Hamiltonian systems are orbitally equivalent.
Example. If k = 1, then we obtain that the Hamiltonians H(1)= 1
A−1r, r2
A2s, s
−Ar, s2
r, r +r, s2
# (67)
and
H¯(1)=As, s
(68)
are orbitally equivalent.
Corollary 6. The Hamiltonian system with Hamiltonian H(1) and the Euler case of the free motion of the rigid body are orbitally equivalent on the orbit Oh, h > 0.
References
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Department of Differential Equations Institut of Mathematics and Informatics Bulgarian Academy of Sciences
Acad. G. Bonchev Str., bl. 8 Sofia 1113
Bulgaria
E-mail address: topalov@math.bas.bg
Primera versi´o rebuda el 30 d’abril de 1999, darrera versi´o rebuda el 23 de setembre de 1999.
